By [1]}, the collection of 1-finely open sets is a topology on \({{{\mathbb {R}}}^n}\) . For a function \(u\in BV({{{\mathbb {R}}}^n})\) , we define the lower approximate limit \(u^\star \)  by setting
\(u_\star (x):=\sup \left\lbrace t\in \overline{{\mathbb {R}}}:\lim _{r\rightarrow 0^+}\frac{\left|B(x, r)\cap \lbrace u<t\rbrace \right|}{\left|B(x, r)\right|}=0\right\rbrace \) 
